Corollary

The group G has a (finite dimensional) faithful linear representation.

Observe first that every element of X(T) is equal to the difference of two dominant weights: more precisely, let $\varpi$ be an element of $X_{++}$ such that $\langle \varpi , K_{\alpha}\rangle >0$ for every simple root $\alpha$; for all $\lambda \in X(T)$ there exists a positive integer $n$ such that $\langle \lambda +n\varpi , K_{\alpha}\rangle \geq 0$ for every simple root $\alpha$, that is (no. 1, Lemma $1$)$\lambda +n\varpi \in X_{++}$.

Consequently, there exists a finite family $(\lambda_i)_{i\in I}$ of elements of $X_{++}$ generating the $\mathbf{Z}$-module X(T). For $i\in I$, let $\tau_i$ be an irreducible representation of G of highest weight $\lambda_i$ (Th. 1); let the representation $\tau$ be the direct sum of the $\tau_i$. By construction the set $P(\tau ,T)$ of weights of $\tau$ (relative to T) generates the $\mathbf{Z}$-module X(T). It now follows from Prop. 6 of §4, no. 3 that the homomorphism $\tau$ is injective, hence the corollary.